Two youths set fire to an old sofa in Tintock Park, in Alloway Gardens, Kirkintilloch, on Friday evening.
The blaze caused extensive damage to safety matting and play equipment in the park.
Firefighters from Kirkintilloch attended the blaze.
Councillor Jack Young says the park is used by children and has condemned the vandals’ actions.
Councillor Young, who represents Kirkintilloch East and Twechar, said: “This is totally irresponsible. Playing with fire at any age is a serious situation.
“Apart from the fact these youngsters could have been seriously injured, it’s the expense involved in calling out the fire brigade, who could have been needed for other more serious fires.
“The council keeps the park in good condition, but vandalism is something we can’t legislate for.
“I’m really disappointed that young people would do this.”
Police are now appealing for information to help them catch those responsible for the fire, which began at around 8.15pm.
One of the youths is described as being 12 to 16-years-old. He was wearing a blue hooded tracksuit.
The second was aged 16 to 20 and was wearing a black hooded tracksuit.
Anyone with information about the fire can contact Kirkintilloch Police Office on 0141 532 4400 or Crimestoppers on 0800 555 111.
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